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INTRODUCTION

To speak of the progress of geometry in descriptive terms, this condensed thesis tries to explain the design process as a deconstruction of geometry. The act of spacial design begins with the most basic geometric principle—the point in space—then moves into a two-dimensional format, which we call drawing, and finally realizes a full, three-dimensional form, which is nothing short of sculpture. As we move from one dimension to the next, we attach artistic concepts of greater complexity to the design process:

(design / design² / design ³ = space / drawing / sculpture)

The key component in this progression is anticipation, which is the acknowledgement that even from the very first stroke of an idea, no matter how basic, the method and stage of design is already looking ahead to the final, three-dimensional artifact.

Finally, this thesis strives to place very specific definitions on the basic language of design, creating a geometrically based lexicon. These definitions appear along the margin and correspond to the text according to immediate relevance.

SPATIALITY

Only a philosophical mind can offer up a concept like one-dimensional spatiality with the serious notion of laying a definition on it, but this is the starting point (quite literally) for the progress of geometry. The significance of a single point in space can't be maligned by excitement over two and three-dimensional concepts as the significance of point-in-space design is not relative, but focal.

Even if a point has no physical value according to geometry, it does have focal value in the language of architecture. For instance, we can preserve point in space design by using columns, each of which is an expression of single point structural dependence, and therefore a theoretical match for the point in space concept. Taken individually, the weight of a structure is focused onto a single point, which in turn yields visual consequences that determine how spatiality functions with that column as a central focus.

A second instance of point in space design preservation is the arrangement of space around a corner. We ascribe a high degree of value to walls, both for their presence as vertical surfaces, and their use for partitioning space. But the corner—the point at which walls end—is equally important. Wherever two walls join together, a natural focal point is created. Depending on the angle of conjunction, two perspective qualities emerge. For a point defined by walls joined at angles of 1° to 179° the result is convex perspective, and for a point defined by walls joined at angles of 181° to 359° the result is concave perspective.

ANTICIPATION

The progress of geometry turns on finite distinctions between spatiality, the drawings that define use of space, and the sculpture that ultimately realizes its form. But the progress of geometry also amits that no design is ever built with a blind eye toward the final artifact, and therefore the process always anticipates its final stages. Anticipation is an inevitability of point in space design; it answers the question, what surrounds this point?

Practically, the progress of geometry works this way: A point in space anticipates two dimensions, even though it expresses none in and of itself. Drawing anticipates three dimensions even though its ability to express restriction of the third dimension is limited. And sculpture realizes the ultimate form.

DRAWING

Because there's such a limited opportunity to express point in space design in a drawing without expressing a second dimension—and indeed it may be impossible not to make such representations implicitly—drawings are inevitably two-dimensional and thus promote the progress of geometry from focal spaciality to relative spaciality. Drawing can express everything about how spaces function relative to one another, or how a single point interacts with its surroundings.

Keeping to the theme of point in space design, a drawing can depict focal spaciality in the way a compass turns around its needle when rendering a circle, or in the way columns and corners are designated within a space. In truth, these actions can occur before any solid features are established, proving that design can be practiced in the most basic, abstract sense. Aesthetically, this blank form of design is so basic it can be called geometric turpitude simply because it has no quantifiable value; it is baseless. This distinction seems purely theoretical, but it's present at the start of every design.

Anticipation plays an important role in drawing. Take for instance a floor plan, which most people consider to be a two-dimensional expression of design. A floor plan not only anticipates the third dimension, but it can also express the third dimension in various ways.

The difference is this: Anticipating the third dimension is acknowledging that to represent a wall by rendering a line infers that the wall has a specific height. Similarly, to render a broken line and label it as a break in a floor or level above or below that of the plan expressed in the drawing also acknowledges some measure of relief in the third dimension. What the floor plan does not do is restrict this third dimension measure. Or, in cases where the restriction is written out, the floor plan is still incapable of expressesing that restriction visibly.

SCULPTURE

Architecture in particular is so heavily geometric in appearance because of the formal considerations that bring it to life. The progress of geometry is a very real phenomenon, else the bulk of civilization wouldn't be housed in assortments of rectilinear cubes. In this sense, the practice of starting at sculpture and working backward leads us to suggest that the progress of geometry is not merely an order of process, but an order of value. To say that point in space design preceeds all other considerations also means it's of a fundamentally higher value. The decisions that are made concerning focal spatiality will affect those that follow without any reservation, unapologetically directing the final shape.

The natural conclusion to draw from this order of value is that architects are lastly sculptors. While this is true, it is also somewhat unfortunate. When an architect first thinks in terms of three-dimensional spaciality and then bends the more basic considerations around this ultimate goal, those who receive the results—the public—are often less likely to accept them. Or the results are polarizing. In turn we might ask if this rejection isn't due to more pedestrian attitidues—a dearth of sophistication?

The progress of geometry is an easy concept to grasp—a natural progression. But the thing about seasoned architects is that they can think forward and backward through the design process. As a result, an architect might feel that the best way to achieve something new is to subvert the natural order of design. So much for being pedestrian, it's more likely that the public recognizes this subversion for what is it, and can go so far as to disagree with the results on a theoretical level even if words aren't there to formalize their sentiments. Whether the maligned target is the chunkiness of Brutalism or the fluid whimsy of Antoni Gaudi, architecture probably stikes discord on a theoretically rich level regardless of the reasons one can express about it. For argument's sake, let's say a glance can be so sophisticated.

But this notion of working backward against the progress of geometry begs the question: Is there a difference between anticipation and subversion? Yes. The difference is in the attention given to the actual design process, separate from the conceptual stage. This distinction speaks to the creative process as it encompasses the less spectacular grunt work of design. To conjure a work of art, it isn't necessary to follow an ordered progression. But, for most any creative mind, once an idea has formed, realizing it in the flesh requires process. Therefore, the progress of geometry applies wherever good design is practiced because the artist or architect is anticipating the ultimate result and considering the best way to get there.

For architecture, the risk and ultimate downfall of subversion is akin to filling a chest with toys, where a shell is conceived, rendered, and restricted in all dimensions, and then stuffed with features and spaces, all of which are relegated to tertiary roles in servitude of the greater sculptural form. This makes for bad design. From the public perspective, distinguishing between the two may not be apparent at a glance, but is likely a judgment to be made once the functionality of a structure is experienced.

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